I think there's a difference between relying on a third party to enforce justice (a concept which I have no problem with), and relying on a third party to choose punishment. If the punishment is exactly in kind (by default), then no choice is required. It's only because the sorts of punishments our society uses are completely different in kind from the initial offense that a choice is even necessary, and that same difference means that there's no "right" answer to the question. There's at least some hope of finding a right answer to how much damage was done to someone after having a TV stolen or being beaten. But even knowing how much damage was done, how could you translate that into a given amount of time in prison in any objective fashion? Is a beating "worth" two years or two months in jail? There's no right answer to that question.

In the system I advocate, actual settlements or punishments might vary widely, but as long as the convict always had the option of taking a similar beating to the one he gave the victim, we'd know that the punishment or settlement couldn't be more than warranted. As long as the victim can refuse any proffered settlement in favor of having the default punishment administered, we know that the settlement can't be less than warranted. The "eye for an eye" standard is just a tool for allowing the convict and victim to negotiate a settlement, and I think it would be rarely fully carried out for anything major, if the victim has the option of accepting some amount of money to forgive some or all of the offense.

but it's also taken as a statement of how you want the world to be.

I don't think people who steal a TV would agree that they want to dissolve property rights. They don't want to have stuff stolen from them, only to get away with it themselves. One of the reasons that law is possible is that even thieves want to have their own property protected, and even burglars want to protect their homes from entry by others.